(15^6)/(9^4) cdot 125^(2)

To simplify the expression \( \frac{15^{6}}{9^{4} \cdot 125^{2}} \), we can start by factoring the numbers in the numerator and denominator.<br /><br />First, let's factor, 9, and 125:<br />- \( 15 = 3 \cdot 5 \)<br />- \( 9 = 3^2 \)<br />- \( 125 = 5^3 \)<br /><br />Now, rewrite the expression using these factorizations:<br />\[ \frac{15^{6}}{9^{4} \cdot 125^{2}} = \frac{(3 \cdot 5)^{6}}{(3^2)^{4} \cdot (5^3)^{2}} \]<br /><br />Next, apply the power rule \((a \cdot b)^n = a^n \cdot b^n\):<br />\[ = \frac{3^6 \cdot 5^6}{3^{8} \cdot 5^6} \]<br /><br />Now, simplify the expression by canceling out common factors in the numerator and denominator:<br />\[ = \frac{3^6 \cdot 5^6}{3^8 \cdot 5^6} = \frac{3^6}{3^8} = \frac{1}{3^2} = \frac{1}{9} \]<br /><br />Therefore, the simplified form of the expression is \( \frac{1}{9} \).